LMFDB - Embedded newform 275.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.30278 q^{2} -1.30278 q^{3} +3.30278 q^{4} -3.00000 q^{6} +4.30278 q^{7} +3.00000 q^{8} -1.30278 q^{9} -1.00000 q^{11} -4.30278 q^{12} +5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -3.00000 q^{18} -1.00000 q^{19} -5.60555 q^{21} -2.30278 q^{22} -3.69722 q^{23} -3.90833 q^{24} +11.5139 q^{26} +5.60555 q^{27} +14.2111 q^{28} -9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} +1.30278 q^{33} -9.00000 q^{34} -4.30278 q^{36} +9.60555 q^{37} -2.30278 q^{38} -6.51388 q^{39} +1.60555 q^{41} -12.9083 q^{42} -7.21110 q^{43} -3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} -0.394449 q^{48} +11.5139 q^{49} +5.09167 q^{51} +16.5139 q^{52} +2.30278 q^{53} +12.9083 q^{54} +12.9083 q^{56} +1.30278 q^{57} -22.8167 q^{58} +0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -5.60555 q^{63} -12.8167 q^{64} +3.00000 q^{66} -4.00000 q^{67} -12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} -3.90833 q^{72} +2.90833 q^{73} +22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -15.0000 q^{78} -0.0916731 q^{79} -3.39445 q^{81} +3.69722 q^{82} +14.5139 q^{83} -18.5139 q^{84} -16.6056 q^{86} +12.9083 q^{87} -3.00000 q^{88} +5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} +5.48612 q^{93} -6.90833 q^{94} +6.90833 q^{96} +11.6972 q^{97} +26.5139 q^{98} +1.30278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{3} + 3 q^{4} - 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} - 2 q^{11} - 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} - 2 q^{19} - 4 q^{21} - q^{22} - 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) $ 2 * q + q^2 + q^3 + 3 * q^4 - 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 - 2 * q^11 - 5 * q^12 + 10 * q^13 + 9 * q^14 - 3 * q^16 + 3 * q^17 - 6 * q^18 - 2 * q^19 - 4 * q^21 - q^22 - 11 * q^23 + 3 * q^24 + 5 * q^26 + 4 * q^27 + 14 * q^28 - 9 * q^29 + 6 * q^31 - 7 * q^32 - q^33 - 18 * q^34 - 5 * q^36 + 12 * q^37 - q^38 + 5 * q^39 - 4 * q^41 - 15 * q^42 - 3 * q^44 + q^46 - 6 * q^47 - 8 * q^48 + 5 * q^49 + 21 * q^51 + 15 * q^52 + q^53 + 15 * q^54 + 15 * q^56 - q^57 - 24 * q^58 - 14 * q^59 - 5 * q^61 - 23 * q^62 - 4 * q^63 - 4 * q^64 + 6 * q^66 - 8 * q^67 - 15 * q^68 - 12 * q^69 + 2 * q^71 + 3 * q^72 - 5 * q^73 + 19 * q^74 - 3 * q^76 - 5 * q^77 - 30 * q^78 - 11 * q^79 - 14 * q^81 + 11 * q^82 + 11 * q^83 - 19 * q^84 - 26 * q^86 + 15 * q^87 - 6 * q^88 + 7 * q^89 + 25 * q^91 - 10 * q^92 + 29 * q^93 - 3 * q^94 + 3 * q^96 + 27 * q^97 + 35 * q^98 - q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	4.30278	1.62630	0.813148	−	0.582057i	$-0.197752\pi$
$7$	4.30278	1.62630	0.813148	+	0.582057i	$0.197752\pi$
$8$	3.00000	1.06066
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.30278	−1.24210
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	9.90833	2.64811
$15$	0	0
$16$	0.302776	0.0756939
$17$	−3.90833	−0.947909	−0.473954	−	0.880549i	$-0.657174\pi$
$17$	−3.90833	−0.947909	−0.473954	+	0.880549i	$0.657174\pi$
$18$	−3.00000	−0.707107
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−5.60555	−1.22323
$22$	−2.30278	−0.490953
$23$	−3.69722	−0.770925	−0.385462	−	0.922724i	$-0.625958\pi$
$23$	−3.69722	−0.770925	−0.385462	+	0.922724i	$0.625958\pi$
$24$	−3.90833	−0.797784
$25$	0	0
$26$	11.5139	2.25806
$27$	5.60555	1.07879
$28$	14.2111	2.68565
$29$	−9.90833	−1.83993	−0.919965	−	0.392000i	$-0.871783\pi$
$29$	−9.90833	−1.83993	−0.919965	+	0.392000i	$0.871783\pi$
$30$	0	0
$31$	−4.21110	−0.756336	−0.378168	−	0.925737i	$-0.623446\pi$
$31$	−4.21110	−0.756336	−0.378168	+	0.925737i	$0.623446\pi$
$32$	−5.30278	−0.937407
$33$	1.30278	0.226784
$34$	−9.00000	−1.54349
$35$	0	0
$36$	−4.30278	−0.717129
$37$	9.60555	1.57914	0.789571	−	0.613659i	$-0.210303\pi$
$37$	9.60555	1.57914	0.789571	+	0.613659i	$0.210303\pi$
$38$	−2.30278	−0.373560
$39$	−6.51388	−1.04306
$40$	0	0
$41$	1.60555	0.250745	0.125372	−	0.992110i	$-0.459987\pi$
$41$	1.60555	0.250745	0.125372	+	0.992110i	$0.459987\pi$
$42$	−12.9083	−1.99180
$43$	−7.21110	−1.09968	−0.549841	−	0.835269i	$-0.685312\pi$
$43$	−7.21110	−1.09968	−0.549841	+	0.835269i	$0.685312\pi$
$44$	−3.30278	−0.497912
$45$	0	0
$46$	−8.51388	−1.25530
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−0.394449	−0.0569338
$49$	11.5139	1.64484
$50$	0	0
$51$	5.09167	0.712977
$52$	16.5139	2.29006
$53$	2.30278	0.316311	0.158155	−	0.987414i	$-0.449445\pi$
$53$	2.30278	0.316311	0.158155	+	0.987414i	$0.449445\pi$
$54$	12.9083	1.75660
$55$	0	0
$56$	12.9083	1.72495
$57$	1.30278	0.172557
$58$	−22.8167	−2.99597
$59$	0.211103	0.0274832	0.0137416	−	0.999906i	$-0.495626\pi$
$59$	0.211103	0.0274832	0.0137416	+	0.999906i	$0.495626\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	−9.69722	−1.23155
$63$	−5.60555	−0.706233
$64$	−12.8167	−1.60208
$65$	0	0
$66$	3.00000	0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−12.9083	−1.56536
$69$	4.81665	0.579857
$70$	0	0
$71$	4.60555	0.546578	0.273289	−	0.961932i	$-0.411888\pi$
$71$	4.60555	0.546578	0.273289	+	0.961932i	$0.411888\pi$
$72$	−3.90833	−0.460601
$73$	2.90833	0.340394	0.170197	−	0.985410i	$-0.445560\pi$
$73$	2.90833	0.340394	0.170197	+	0.985410i	$0.445560\pi$
$74$	22.1194	2.57133
$75$	0	0
$76$	−3.30278	−0.378854
$77$	−4.30278	−0.490347
$78$	−15.0000	−1.69842
$79$	−0.0916731	−0.0103140	−0.00515701	−	0.999987i	$-0.501642\pi$
$79$	−0.0916731	−0.0103140	−0.00515701	+	0.999987i	$0.501642\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	3.69722	0.408290
$83$	14.5139	1.59311	0.796553	−	0.604569i	$-0.206655\pi$
$83$	14.5139	1.59311	0.796553	+	0.604569i	$0.206655\pi$
$84$	−18.5139	−2.02003
$85$	0	0
$86$	−16.6056	−1.79062
$87$	12.9083	1.38392
$88$	−3.00000	−0.319801
$89$	5.30278	0.562093	0.281047	−	0.959694i	$-0.409318\pi$
$89$	5.30278	0.562093	0.281047	+	0.959694i	$0.409318\pi$
$90$	0	0
$91$	21.5139	2.25527
$92$	−12.2111	−1.27310
$93$	5.48612	0.568884
$94$	−6.90833	−0.712540
$95$	0	0
$96$	6.90833	0.705078
$97$	11.6972	1.18767	0.593837	−	0.804586i	$-0.297613\pi$
$97$	11.6972	1.18767	0.593837	+	0.804586i	$0.297613\pi$
$98$	26.5139	2.67831
$99$	1.30278	0.130934
$100$	0	0
$101$	17.5139	1.74270	0.871348	−	0.490666i	$-0.163246\pi$
$101$	17.5139	1.74270	0.871348	+	0.490666i	$0.163246\pi$
$102$	11.7250	1.16095
$103$	−7.90833	−0.779231	−0.389615	−	0.920978i	$-0.627392\pi$
$103$	−7.90833	−0.779231	−0.389615	+	0.920978i	$0.627392\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	5.30278	0.515051
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	18.5139	1.78150
$109$	−6.51388	−0.623916	−0.311958	−	0.950096i	$-0.600985\pi$
$109$	−6.51388	−0.623916	−0.311958	+	0.950096i	$0.600985\pi$
$110$	0	0
$111$	−12.5139	−1.18776
$112$	1.30278	0.123101
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−32.7250	−3.03844
$117$	−6.51388	−0.602208
$118$	0.486122	0.0447511
$119$	−16.8167	−1.54158
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.69722	0.606338
$123$	−2.09167	−0.188600
$124$	−13.9083	−1.24900
$125$	0	0
$126$	−12.9083	−1.14997
$127$	−17.1194	−1.51910	−0.759552	−	0.650447i	$-0.774582\pi$
$127$	−17.1194	−1.51910	−0.759552	+	0.650447i	$0.774582\pi$
$128$	−18.9083	−1.67128
$129$	9.39445	0.827135
$130$	0	0
$131$	0.908327	0.0793609	0.0396804	−	0.999212i	$-0.487366\pi$
$131$	0.908327	0.0793609	0.0396804	+	0.999212i	$0.487366\pi$
$132$	4.30278	0.374509
$133$	−4.30278	−0.373098
$134$	−9.21110	−0.795718
$135$	0	0
$136$	−11.7250	−1.00541
$137$	−2.09167	−0.178704	−0.0893518	−	0.996000i	$-0.528480\pi$
$137$	−2.09167	−0.178704	−0.0893518	+	0.996000i	$0.528480\pi$
$138$	11.0917	0.944186
$139$	8.21110	0.696457	0.348228	−	0.937410i	$-0.386783\pi$
$139$	8.21110	0.696457	0.348228	+	0.937410i	$0.386783\pi$
$140$	0	0
$141$	3.90833	0.329141
$142$	10.6056	0.889998
$143$	−5.00000	−0.418121
$144$	−0.394449	−0.0328707
$145$	0	0
$146$	6.69722	0.554266
$147$	−15.0000	−1.23718
$148$	31.7250	2.60778
$149$	2.78890	0.228475	0.114238	−	0.993453i	$-0.463557\pi$
$149$	2.78890	0.228475	0.114238	+	0.993453i	$0.463557\pi$
$150$	0	0
$151$	−20.8167	−1.69404	−0.847018	−	0.531565i	$-0.821604\pi$
$151$	−20.8167	−1.69404	−0.847018	+	0.531565i	$0.821604\pi$
$152$	−3.00000	−0.243332
$153$	5.09167	0.411637
$154$	−9.90833	−0.798436
$155$	0	0
$156$	−21.5139	−1.72249
$157$	4.78890	0.382196	0.191098	−	0.981571i	$-0.438795\pi$
$157$	4.78890	0.382196	0.191098	+	0.981571i	$0.438795\pi$
$158$	−0.211103	−0.0167944
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−15.9083	−1.25375
$162$	−7.81665	−0.614134
$163$	5.69722	0.446241	0.223121	−	0.974791i	$-0.428376\pi$
$163$	5.69722	0.446241	0.223121	+	0.974791i	$0.428376\pi$
$164$	5.30278	0.414077
$165$	0	0
$166$	33.4222	2.59407
$167$	−15.4222	−1.19341	−0.596703	−	0.802462i	$-0.703523\pi$
$167$	−15.4222	−1.19341	−0.596703	+	0.802462i	$0.703523\pi$
$168$	−16.8167	−1.29743
$169$	12.0000	0.923077
$170$	0	0
$171$	1.30278	0.0996257
$172$	−23.8167	−1.81600
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	29.7250	2.25344
$175$	0	0
$176$	−0.302776	−0.0228226
$177$	−0.275019	−0.0206717
$178$	12.2111	0.915261
$179$	−5.51388	−0.412127	−0.206063	−	0.978539i	$-0.566065\pi$
$179$	−5.51388	−0.412127	−0.206063	+	0.978539i	$0.566065\pi$
$180$	0	0
$181$	−9.09167	−0.675779	−0.337889	−	0.941186i	$-0.609713\pi$
$181$	−9.09167	−0.675779	−0.337889	+	0.941186i	$0.609713\pi$
$182$	49.5416	3.67227
$183$	−3.78890	−0.280083
$184$	−11.0917	−0.817689
$185$	0	0
$186$	12.6333	0.926319
$187$	3.90833	0.285805
$188$	−9.90833	−0.722639
$189$	24.1194	1.75443
$190$	0	0
$191$	6.69722	0.484594	0.242297	−	0.970202i	$-0.422099\pi$
$191$	6.69722	0.484594	0.242297	+	0.970202i	$0.422099\pi$
$192$	16.6972	1.20502
$193$	−1.21110	−0.0871771	−0.0435885	−	0.999050i	$-0.513879\pi$
$193$	−1.21110	−0.0871771	−0.0435885	+	0.999050i	$0.513879\pi$
$194$	26.9361	1.93390
$195$	0	0
$196$	38.0278	2.71627
$197$	9.69722	0.690899	0.345449	−	0.938437i	$-0.387727\pi$
$197$	9.69722	0.690899	0.345449	+	0.938437i	$0.387727\pi$
$198$	3.00000	0.213201
$199$	−24.5139	−1.73774	−0.868871	−	0.495038i	$-0.835154\pi$
$199$	−24.5139	−1.73774	−0.868871	+	0.495038i	$0.835154\pi$
$200$	0	0
$201$	5.21110	0.367563
$202$	40.3305	2.83765
$203$	−42.6333	−2.99227
$204$	16.8167	1.17740
$205$	0	0
$206$	−18.2111	−1.26883
$207$	4.81665	0.334781
$208$	1.51388	0.104969
$209$	1.00000	0.0691714
$210$	0	0
$211$	25.2389	1.73751	0.868757	−	0.495238i	$-0.164919\pi$
$211$	25.2389	1.73751	0.868757	+	0.495238i	$0.164919\pi$
$212$	7.60555	0.522351
$213$	−6.00000	−0.411113
$214$	−6.90833	−0.472244
$215$	0	0
$216$	16.8167	1.14423
$217$	−18.1194	−1.23003
$218$	−15.0000	−1.01593
$219$	−3.78890	−0.256030
$220$	0	0
$221$	−19.5416	−1.31451
$222$	−28.8167	−1.93405
$223$	20.6333	1.38171	0.690854	−	0.722994i	$-0.257235\pi$
$223$	20.6333	1.38171	0.690854	+	0.722994i	$0.257235\pi$
$224$	−22.8167	−1.52450
$225$	0	0
$226$	24.9083	1.65688
$227$	5.30278	0.351958	0.175979	−	0.984394i	$-0.443691\pi$
$227$	5.30278	0.351958	0.175979	+	0.984394i	$0.443691\pi$
$228$	4.30278	0.284958
$229$	13.7250	0.906972	0.453486	−	0.891263i	$-0.350180\pi$
$229$	13.7250	0.906972	0.453486	+	0.891263i	$0.350180\pi$
$230$	0	0
$231$	5.60555	0.368818
$232$	−29.7250	−1.95154
$233$	5.09167	0.333567	0.166783	−	0.985994i	$-0.446662\pi$
$233$	5.09167	0.333567	0.166783	+	0.985994i	$0.446662\pi$
$234$	−15.0000	−0.980581
$235$	0	0
$236$	0.697224	0.0453854
$237$	0.119429	0.00775778
$238$	−38.7250	−2.51017
$239$	−4.11943	−0.266464	−0.133232	−	0.991085i	$-0.542535\pi$
$239$	−4.11943	−0.266464	−0.133232	+	0.991085i	$0.542535\pi$
$240$	0	0
$241$	−24.9361	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$241$	−24.9361	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$242$	2.30278	0.148028
$243$	−12.3944	−0.795104
$244$	9.60555	0.614932
$245$	0	0
$246$	−4.81665	−0.307099
$247$	−5.00000	−0.318142
$248$	−12.6333	−0.802216
$249$	−18.9083	−1.19827
$250$	0	0
$251$	−3.90833	−0.246691	−0.123346	−	0.992364i	$-0.539362\pi$
$251$	−3.90833	−0.246691	−0.123346	+	0.992364i	$0.539362\pi$
$252$	−18.5139	−1.16626
$253$	3.69722	0.232443
$254$	−39.4222	−2.47357
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	21.6333	1.34683
$259$	41.3305	2.56815
$260$	0	0
$261$	12.9083	0.799005
$262$	2.09167	0.129224
$263$	−1.18335	−0.0729683	−0.0364841	−	0.999334i	$-0.511616\pi$
$263$	−1.18335	−0.0729683	−0.0364841	+	0.999334i	$0.511616\pi$
$264$	3.90833	0.240541
$265$	0	0
$266$	−9.90833	−0.607519
$267$	−6.90833	−0.422783
$268$	−13.2111	−0.806997
$269$	−23.7250	−1.44654	−0.723269	−	0.690567i	$-0.757361\pi$
$269$	−23.7250	−1.44654	−0.723269	+	0.690567i	$0.757361\pi$
$270$	0	0
$271$	14.2111	0.863263	0.431632	−	0.902050i	$-0.357938\pi$
$271$	14.2111	0.863263	0.431632	+	0.902050i	$0.357938\pi$
$272$	−1.18335	−0.0717509
$273$	−28.0278	−1.69632
$274$	−4.81665	−0.290985
$275$	0	0
$276$	15.9083	0.957569
$277$	21.6056	1.29815	0.649076	−	0.760724i	$-0.275156\pi$
$277$	21.6056	1.29815	0.649076	+	0.760724i	$0.275156\pi$
$278$	18.9083	1.13405
$279$	5.48612	0.328446
$280$	0	0
$281$	−22.8167	−1.36113	−0.680564	−	0.732689i	$-0.738265\pi$
$281$	−22.8167	−1.36113	−0.680564	+	0.732689i	$0.738265\pi$
$282$	9.00000	0.535942
$283$	2.69722	0.160333	0.0801667	−	0.996781i	$-0.474455\pi$
$283$	2.69722	0.160333	0.0801667	+	0.996781i	$0.474455\pi$
$284$	15.2111	0.902613
$285$	0	0
$286$	−11.5139	−0.680830
$287$	6.90833	0.407786
$288$	6.90833	0.407077
$289$	−1.72498	−0.101469
$290$	0	0
$291$	−15.2389	−0.893318
$292$	9.60555	0.562122
$293$	15.2111	0.888642	0.444321	−	0.895868i	$-0.353445\pi$
$293$	15.2111	0.888642	0.444321	+	0.895868i	$0.353445\pi$
$294$	−34.5416	−2.01451
$295$	0	0
$296$	28.8167	1.67493
$297$	−5.60555	−0.325267
$298$	6.42221	0.372028
$299$	−18.4861	−1.06908
$300$	0	0
$301$	−31.0278	−1.78841
$302$	−47.9361	−2.75841
$303$	−22.8167	−1.31078
$304$	−0.302776	−0.0173654
$305$	0	0
$306$	11.7250	0.670273
$307$	−6.09167	−0.347670	−0.173835	−	0.984775i	$-0.555616\pi$
$307$	−6.09167	−0.347670	−0.173835	+	0.984775i	$0.555616\pi$
$308$	−14.2111	−0.809753
$309$	10.3028	0.586104
$310$	0	0
$311$	−16.8167	−0.953585	−0.476792	−	0.879016i	$-0.658201\pi$
$311$	−16.8167	−0.953585	−0.476792	+	0.879016i	$0.658201\pi$
$312$	−19.5416	−1.10633
$313$	21.8167	1.23315	0.616575	−	0.787296i	$-0.288520\pi$
$313$	21.8167	1.23315	0.616575	+	0.787296i	$0.288520\pi$
$314$	11.0278	0.622332
$315$	0	0
$316$	−0.302776	−0.0170325
$317$	9.90833	0.556507	0.278254	−	0.960508i	$-0.410244\pi$
$317$	9.90833	0.556507	0.278254	+	0.960508i	$0.410244\pi$
$318$	−6.90833	−0.387400
$319$	9.90833	0.554760
$320$	0	0
$321$	3.90833	0.218142
$322$	−36.6333	−2.04149
$323$	3.90833	0.217465
$324$	−11.2111	−0.622839
$325$	0	0
$326$	13.1194	0.726618
$327$	8.48612	0.469284
$328$	4.81665	0.265955
$329$	−12.9083	−0.711659
$330$	0	0
$331$	−14.3944	−0.791190	−0.395595	−	0.918425i	$-0.629462\pi$
$331$	−14.3944	−0.791190	−0.395595	+	0.918425i	$0.629462\pi$
$332$	47.9361	2.63083
$333$	−12.5139	−0.685756
$334$	−35.5139	−1.94323
$335$	0	0
$336$	−1.69722	−0.0925912
$337$	26.8444	1.46231	0.731154	−	0.682212i	$-0.238982\pi$
$337$	26.8444	1.46231	0.731154	+	0.682212i	$0.238982\pi$
$338$	27.6333	1.50305
$339$	−14.0917	−0.765355
$340$	0	0
$341$	4.21110	0.228044
$342$	3.00000	0.162221
$343$	19.4222	1.04870
$344$	−21.6333	−1.16639
$345$	0	0
$346$	38.7250	2.08187
$347$	5.51388	0.296000	0.148000	−	0.988987i	$-0.452716\pi$
$347$	5.51388	0.296000	0.148000	+	0.988987i	$0.452716\pi$
$348$	42.6333	2.28539
$349$	−26.8167	−1.43546	−0.717731	−	0.696320i	$-0.754819\pi$
$349$	−26.8167	−1.43546	−0.717731	+	0.696320i	$0.754819\pi$
$350$	0	0
$351$	28.0278	1.49601
$352$	5.30278	0.282639
$353$	−24.6333	−1.31110	−0.655549	−	0.755152i	$-0.727563\pi$
$353$	−24.6333	−1.31110	−0.655549	+	0.755152i	$0.727563\pi$
$354$	−0.633308	−0.0336599
$355$	0	0
$356$	17.5139	0.928234
$357$	21.9083	1.15951
$358$	−12.6972	−0.671069
$359$	15.2111	0.802811	0.401406	−	0.915900i	$-0.368522\pi$
$359$	15.2111	0.802811	0.401406	+	0.915900i	$0.368522\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−20.9361	−1.10038
$363$	−1.30278	−0.0683780
$364$	71.0555	3.72432
$365$	0	0
$366$	−8.72498	−0.456062
$367$	−24.3028	−1.26859	−0.634297	−	0.773089i	$-0.718710\pi$
$367$	−24.3028	−1.26859	−0.634297	+	0.773089i	$0.718710\pi$
$368$	−1.11943	−0.0583543
$369$	−2.09167	−0.108888
$370$	0	0
$371$	9.90833	0.514415
$372$	18.1194	0.939449
$373$	−1.42221	−0.0736390	−0.0368195	−	0.999322i	$-0.511723\pi$
$373$	−1.42221	−0.0736390	−0.0368195	+	0.999322i	$0.511723\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	−49.5416	−2.55152
$378$	55.5416	2.85675
$379$	24.8167	1.27475	0.637373	−	0.770555i	$-0.280021\pi$
$379$	24.8167	1.27475	0.637373	+	0.770555i	$0.280021\pi$
$380$	0	0
$381$	22.3028	1.14261
$382$	15.4222	0.789069
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	24.6333	1.25706
$385$	0	0
$386$	−2.78890	−0.141951
$387$	9.39445	0.477547
$388$	38.6333	1.96131
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	14.4500	0.730766
$392$	34.5416	1.74462
$393$	−1.18335	−0.0596919
$394$	22.3305	1.12500
$395$	0	0
$396$	4.30278	0.216223
$397$	25.3028	1.26991	0.634955	−	0.772549i	$-0.281019\pi$
$397$	25.3028	1.26991	0.634955	+	0.772549i	$0.281019\pi$
$398$	−56.4500	−2.82958
$399$	5.60555	0.280629
$400$	0	0
$401$	−27.2111	−1.35886	−0.679429	−	0.733741i	$-0.737772\pi$
$401$	−27.2111	−1.35886	−0.679429	+	0.733741i	$0.737772\pi$
$402$	12.0000	0.598506
$403$	−21.0555	−1.04885
$404$	57.8444	2.87787
$405$	0	0
$406$	−98.1749	−4.87234
$407$	−9.60555	−0.476129
$408$	15.2750	0.756226
$409$	8.21110	0.406013	0.203006	−	0.979177i	$-0.434929\pi$
$409$	8.21110	0.406013	0.203006	+	0.979177i	$0.434929\pi$
$410$	0	0
$411$	2.72498	0.134413
$412$	−26.1194	−1.28681
$413$	0.908327	0.0446958
$414$	11.0917	0.545126
$415$	0	0
$416$	−26.5139	−1.29995
$417$	−10.6972	−0.523845
$418$	2.30278	0.112632
$419$	13.6056	0.664675	0.332337	−	0.943161i	$-0.392163\pi$
$419$	13.6056	0.664675	0.332337	+	0.943161i	$0.392163\pi$
$420$	0	0
$421$	4.30278	0.209704	0.104852	−	0.994488i	$-0.466563\pi$
$421$	4.30278	0.209704	0.104852	+	0.994488i	$0.466563\pi$
$422$	58.1194	2.82921
$423$	3.90833	0.190029
$424$	6.90833	0.335498
$425$	0	0
$426$	−13.8167	−0.669419
$427$	12.5139	0.605589
$428$	−9.90833	−0.478937
$429$	6.51388	0.314493
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	1.69722	0.0816577
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	−41.7250	−2.00286
$435$	0	0
$436$	−21.5139	−1.03033
$437$	3.69722	0.176862
$438$	−8.72498	−0.416896
$439$	20.6972	0.987825	0.493912	−	0.869512i	$-0.335566\pi$
$439$	20.6972	0.987825	0.493912	+	0.869512i	$0.335566\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	−45.0000	−2.14043
$443$	−1.39445	−0.0662523	−0.0331261	−	0.999451i	$-0.510546\pi$
$443$	−1.39445	−0.0662523	−0.0331261	+	0.999451i	$0.510546\pi$
$444$	−41.3305	−1.96146
$445$	0	0
$446$	47.5139	2.24985
$447$	−3.63331	−0.171850
$448$	−55.1472	−2.60546
$449$	41.5139	1.95916	0.979581	−	0.201052i	$-0.0644361\pi$
$449$	41.5139	1.95916	0.979581	+	0.201052i	$0.0644361\pi$
$450$	0	0
$451$	−1.60555	−0.0756025
$452$	35.7250	1.68036
$453$	27.1194	1.27418
$454$	12.2111	0.573095
$455$	0	0
$456$	3.90833	0.183024
$457$	−24.3028	−1.13684	−0.568418	−	0.822740i	$-0.692444\pi$
$457$	−24.3028	−1.13684	−0.568418	+	0.822740i	$0.692444\pi$
$458$	31.6056	1.47683
$459$	−21.9083	−1.02259
$460$	0	0
$461$	17.7889	0.828512	0.414256	−	0.910161i	$-0.364042\pi$
$461$	17.7889	0.828512	0.414256	+	0.910161i	$0.364042\pi$
$462$	12.9083	0.600550
$463$	26.2111	1.21813	0.609067	−	0.793119i	$-0.291544\pi$
$463$	26.2111	1.21813	0.609067	+	0.793119i	$0.291544\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	11.7250	0.543149
$467$	24.6333	1.13989	0.569947	−	0.821682i	$-0.306964\pi$
$467$	24.6333	1.13989	0.569947	+	0.821682i	$0.306964\pi$
$468$	−21.5139	−0.994479
$469$	−17.2111	−0.794735
$470$	0	0
$471$	−6.23886	−0.287471
$472$	0.633308	0.0291503
$473$	7.21110	0.331567
$474$	0.275019	0.0126321
$475$	0	0
$476$	−55.5416	−2.54575
$477$	−3.00000	−0.137361
$478$	−9.48612	−0.433885
$479$	−13.1833	−0.602362	−0.301181	−	0.953567i	$-0.597381\pi$
$479$	−13.1833	−0.602362	−0.301181	+	0.953567i	$0.597381\pi$
$480$	0	0
$481$	48.0278	2.18988
$482$	−57.4222	−2.61551
$483$	20.7250	0.943019
$484$	3.30278	0.150126
$485$	0	0
$486$	−28.5416	−1.29467
$487$	−10.2111	−0.462709	−0.231355	−	0.972869i	$-0.574316\pi$
$487$	−10.2111	−0.462709	−0.231355	+	0.972869i	$0.574316\pi$
$488$	8.72498	0.394961
$489$	−7.42221	−0.335644
$490$	0	0
$491$	−24.2111	−1.09263	−0.546316	−	0.837579i	$-0.683970\pi$
$491$	−24.2111	−1.09263	−0.546316	+	0.837579i	$0.683970\pi$
$492$	−6.90833	−0.311451
$493$	38.7250	1.74409
$494$	−11.5139	−0.518034
$495$	0	0
$496$	−1.27502	−0.0572501
$497$	19.8167	0.888898
$498$	−43.5416	−1.95115
$499$	−21.5139	−0.963093	−0.481547	−	0.876420i	$-0.659925\pi$
$499$	−21.5139	−0.963093	−0.481547	+	0.876420i	$0.659925\pi$
$500$	0	0
$501$	20.0917	0.897630
$502$	−9.00000	−0.401690
$503$	16.6056	0.740405	0.370202	−	0.928951i	$-0.379288\pi$
$503$	16.6056	0.740405	0.370202	+	0.928951i	$0.379288\pi$
$504$	−16.8167	−0.749073
$505$	0	0
$506$	8.51388	0.378488
$507$	−15.6333	−0.694300
$508$	−56.5416	−2.50863
$509$	−26.3028	−1.16585	−0.582925	−	0.812526i	$-0.698092\pi$
$509$	−26.3028	−1.16585	−0.582925	+	0.812526i	$0.698092\pi$
$510$	0	0
$511$	12.5139	0.553581
$512$	−3.42221	−0.151242
$513$	−5.60555	−0.247491
$514$	−41.4500	−1.82828
$515$	0	0
$516$	31.0278	1.36592
$517$	3.00000	0.131940
$518$	95.1749	4.18175
$519$	−21.9083	−0.961669
$520$	0	0
$521$	23.4500	1.02736	0.513681	−	0.857981i	$-0.328282\pi$
$521$	23.4500	1.02736	0.513681	+	0.857981i	$0.328282\pi$
$522$	29.7250	1.30103
$523$	−3.57779	−0.156446	−0.0782230	−	0.996936i	$-0.524925\pi$
$523$	−3.57779	−0.156446	−0.0782230	+	0.996936i	$0.524925\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	−2.72498	−0.118815
$527$	16.4584	0.716938
$528$	0.394449	0.0171662
$529$	−9.33053	−0.405675
$530$	0	0
$531$	−0.275019	−0.0119348
$532$	−14.2111	−0.616129
$533$	8.02776	0.347721
$534$	−15.9083	−0.688421
$535$	0	0
$536$	−12.0000	−0.518321
$537$	7.18335	0.309984
$538$	−54.6333	−2.35541
$539$	−11.5139	−0.495938
$540$	0	0
$541$	−6.72498	−0.289130	−0.144565	−	0.989495i	$-0.546178\pi$
$541$	−6.72498	−0.289130	−0.144565	+	0.989495i	$0.546178\pi$
$542$	32.7250	1.40566
$543$	11.8444	0.508292
$544$	20.7250	0.888576
$545$	0	0
$546$	−64.5416	−2.76213
$547$	18.1194	0.774731	0.387365	−	0.921926i	$-0.373385\pi$
$547$	18.1194	0.774731	0.387365	+	0.921926i	$0.373385\pi$
$548$	−6.90833	−0.295109
$549$	−3.78890	−0.161706
$550$	0	0
$551$	9.90833	0.422109
$552$	14.4500	0.615031
$553$	−0.394449	−0.0167737
$554$	49.7527	2.11379
$555$	0	0
$556$	27.1194	1.15012
$557$	−9.42221	−0.399232	−0.199616	−	0.979874i	$-0.563969\pi$
$557$	−9.42221	−0.399232	−0.199616	+	0.979874i	$0.563969\pi$
$558$	12.6333	0.534811
$559$	−36.0555	−1.52499
$560$	0	0
$561$	−5.09167	−0.214971
$562$	−52.5416	−2.21634
$563$	18.9083	0.796891	0.398445	−	0.917192i	$-0.369550\pi$
$563$	18.9083	0.796891	0.398445	+	0.917192i	$0.369550\pi$
$564$	12.9083	0.543539
$565$	0	0
$566$	6.21110	0.261072
$567$	−14.6056	−0.613375
$568$	13.8167	0.579734
$569$	15.1472	0.635003	0.317502	−	0.948258i	$-0.397156\pi$
$569$	15.1472	0.635003	0.317502	+	0.948258i	$0.397156\pi$
$570$	0	0
$571$	−17.3305	−0.725260	−0.362630	−	0.931933i	$-0.618121\pi$
$571$	−17.3305	−0.725260	−0.362630	+	0.931933i	$0.618121\pi$
$572$	−16.5139	−0.690480
$573$	−8.72498	−0.364491
$574$	15.9083	0.664001
$575$	0	0
$576$	16.6972	0.695718
$577$	−31.3583	−1.30546	−0.652731	−	0.757589i	$-0.726377\pi$
$577$	−31.3583	−1.30546	−0.652731	+	0.757589i	$0.726377\pi$
$578$	−3.97224	−0.165224
$579$	1.57779	0.0655709
$580$	0	0
$581$	62.4500	2.59086
$582$	−35.0917	−1.45460
$583$	−2.30278	−0.0953712
$584$	8.72498	0.361042
$585$	0	0
$586$	35.0278	1.44698
$587$	−37.5416	−1.54951	−0.774755	−	0.632262i	$-0.782127\pi$
$587$	−37.5416	−1.54951	−0.774755	+	0.632262i	$0.782127\pi$
$588$	−49.5416	−2.04306
$589$	4.21110	0.173515
$590$	0	0
$591$	−12.6333	−0.519665
$592$	2.90833	0.119531
$593$	−13.6056	−0.558713	−0.279357	−	0.960187i	$-0.590121\pi$
$593$	−13.6056	−0.558713	−0.279357	+	0.960187i	$0.590121\pi$
$594$	−12.9083	−0.529635
$595$	0	0
$596$	9.21110	0.377301
$597$	31.9361	1.30706
$598$	−42.5694	−1.74079
$599$	−14.0917	−0.575770	−0.287885	−	0.957665i	$-0.592952\pi$
$599$	−14.0917	−0.575770	−0.287885	+	0.957665i	$0.592952\pi$
$600$	0	0
$601$	8.90833	0.363378	0.181689	−	0.983356i	$-0.441844\pi$
$601$	8.90833	0.363378	0.181689	+	0.983356i	$0.441844\pi$
$602$	−71.4500	−2.91208
$603$	5.21110	0.212213
$604$	−68.7527	−2.79751
$605$	0	0
$606$	−52.5416	−2.13436
$607$	−7.21110	−0.292690	−0.146345	−	0.989234i	$-0.546751\pi$
$607$	−7.21110	−0.292690	−0.146345	+	0.989234i	$0.546751\pi$
$608$	5.30278	0.215056
$609$	55.5416	2.25066
$610$	0	0
$611$	−15.0000	−0.606835
$612$	16.8167	0.679773
$613$	−41.1194	−1.66080	−0.830399	−	0.557169i	$-0.811888\pi$
$613$	−41.1194	−1.66080	−0.830399	+	0.557169i	$0.811888\pi$
$614$	−14.0278	−0.566114
$615$	0	0
$616$	−12.9083	−0.520091
$617$	10.6056	0.426963	0.213482	−	0.976947i	$-0.431520\pi$
$617$	10.6056	0.426963	0.213482	+	0.976947i	$0.431520\pi$
$618$	23.7250	0.954359
$619$	17.4222	0.700258	0.350129	−	0.936702i	$-0.386138\pi$
$619$	17.4222	0.700258	0.350129	+	0.936702i	$0.386138\pi$
$620$	0	0
$621$	−20.7250	−0.831665
$622$	−38.7250	−1.55273
$623$	22.8167	0.914130
$624$	−1.97224	−0.0789529
$625$	0	0
$626$	50.2389	2.00795
$627$	−1.30278	−0.0520278
$628$	15.8167	0.631153
$629$	−37.5416	−1.49688
$630$	0	0
$631$	−39.9361	−1.58983	−0.794915	−	0.606721i	$-0.792485\pi$
$631$	−39.9361	−1.58983	−0.794915	+	0.606721i	$0.792485\pi$
$632$	−0.275019	−0.0109397
$633$	−32.8806	−1.30689
$634$	22.8167	0.906165
$635$	0	0
$636$	−9.90833	−0.392891
$637$	57.5694	2.28098
$638$	22.8167	0.903320
$639$	−6.00000	−0.237356
$640$	0	0
$641$	42.2111	1.66724	0.833619	−	0.552340i	$-0.186265\pi$
$641$	42.2111	1.66724	0.833619	+	0.552340i	$0.186265\pi$
$642$	9.00000	0.355202
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	−52.5416	−2.07043
$645$	0	0
$646$	9.00000	0.354100
$647$	17.2389	0.677729	0.338865	−	0.940835i	$-0.389957\pi$
$647$	17.2389	0.677729	0.338865	+	0.940835i	$0.389957\pi$
$648$	−10.1833	−0.400040
$649$	−0.211103	−0.00828650
$650$	0	0
$651$	23.6056	0.925174
$652$	18.8167	0.736917
$653$	−19.1194	−0.748201	−0.374101	−	0.927388i	$-0.622049\pi$
$653$	−19.1194	−0.748201	−0.374101	+	0.927388i	$0.622049\pi$
$654$	19.5416	0.764138
$655$	0	0
$656$	0.486122	0.0189799
$657$	−3.78890	−0.147819
$658$	−29.7250	−1.15880
$659$	−20.0917	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$659$	−20.0917	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$660$	0	0
$661$	12.8167	0.498510	0.249255	−	0.968438i	$-0.419814\pi$
$661$	12.8167	0.498510	0.249255	+	0.968438i	$0.419814\pi$
$662$	−33.1472	−1.28830
$663$	25.4584	0.988721
$664$	43.5416	1.68974
$665$	0	0
$666$	−28.8167	−1.11662
$667$	36.6333	1.41845
$668$	−50.9361	−1.97078
$669$	−26.8806	−1.03926
$670$	0	0
$671$	−2.90833	−0.112275
$672$	29.7250	1.14667
$673$	−6.02776	−0.232353	−0.116176	−	0.993229i	$-0.537064\pi$
$673$	−6.02776	−0.232353	−0.116176	+	0.993229i	$0.537064\pi$
$674$	61.8167	2.38109
$675$	0	0
$676$	39.6333	1.52436
$677$	−26.2389	−1.00844	−0.504221	−	0.863575i	$-0.668220\pi$
$677$	−26.2389	−1.00844	−0.504221	+	0.863575i	$0.668220\pi$
$678$	−32.4500	−1.24623
$679$	50.3305	1.93151
$680$	0	0
$681$	−6.90833	−0.264728
$682$	9.69722	0.371326
$683$	−9.84441	−0.376686	−0.188343	−	0.982103i	$-0.560312\pi$
$683$	−9.84441	−0.376686	−0.188343	+	0.982103i	$0.560312\pi$
$684$	4.30278	0.164521
$685$	0	0
$686$	44.7250	1.70761
$687$	−17.8806	−0.682186
$688$	−2.18335	−0.0832393
$689$	11.5139	0.438644
$690$	0	0
$691$	−26.5416	−1.00969	−0.504846	−	0.863210i	$-0.668451\pi$
$691$	−26.5416	−1.00969	−0.504846	+	0.863210i	$0.668451\pi$
$692$	55.5416	2.11138
$693$	5.60555	0.212937
$694$	12.6972	0.481980
$695$	0	0
$696$	38.7250	1.46787
$697$	−6.27502	−0.237683
$698$	−61.7527	−2.33738
$699$	−6.63331	−0.250895
$700$	0	0
$701$	−26.7889	−1.01180	−0.505901	−	0.862591i	$-0.668840\pi$
$701$	−26.7889	−1.01180	−0.505901	+	0.862591i	$0.668840\pi$
$702$	64.5416	2.43597
$703$	−9.60555	−0.362280
$704$	12.8167	0.483046
$705$	0	0
$706$	−56.7250	−2.13487
$707$	75.3583	2.83414
$708$	−0.908327	−0.0341370
$709$	11.6333	0.436898	0.218449	−	0.975848i	$-0.429900\pi$
$709$	11.6333	0.436898	0.218449	+	0.975848i	$0.429900\pi$
$710$	0	0
$711$	0.119429	0.00447895
$712$	15.9083	0.596190
$713$	15.5694	0.583078
$714$	50.4500	1.88804
$715$	0	0
$716$	−18.2111	−0.680581
$717$	5.36669	0.200423
$718$	35.0278	1.30722
$719$	28.8167	1.07468	0.537340	−	0.843366i	$-0.319429\pi$
$719$	28.8167	1.07468	0.537340	+	0.843366i	$0.319429\pi$
$720$	0	0
$721$	−34.0278	−1.26726
$722$	−41.4500	−1.54261
$723$	32.4861	1.20817
$724$	−30.0278	−1.11597
$725$	0	0
$726$	−3.00000	−0.111340
$727$	0.330532	0.0122588	0.00612938	−	0.999981i	$-0.498049\pi$
$727$	0.330532	0.0122588	0.00612938	+	0.999981i	$0.498049\pi$
$728$	64.5416	2.39207
$729$	26.3305	0.975205
$730$	0	0
$731$	28.1833	1.04240
$732$	−12.5139	−0.462526
$733$	12.3944	0.457799	0.228900	−	0.973450i	$-0.426487\pi$
$733$	12.3944	0.457799	0.228900	+	0.973450i	$0.426487\pi$
$734$	−55.9638	−2.06566
$735$	0	0
$736$	19.6056	0.722670
$737$	4.00000	0.147342
$738$	−4.81665	−0.177303
$739$	9.88057	0.363463	0.181731	−	0.983348i	$-0.441830\pi$
$739$	9.88057	0.363463	0.181731	+	0.983348i	$0.441830\pi$
$740$	0	0
$741$	6.51388	0.239293
$742$	22.8167	0.837626
$743$	44.3028	1.62531	0.812656	−	0.582744i	$-0.198021\pi$
$743$	44.3028	1.62531	0.812656	+	0.582744i	$0.198021\pi$
$744$	16.4584	0.603393
$745$	0	0
$746$	−3.27502	−0.119907
$747$	−18.9083	−0.691820
$748$	12.9083	0.471975
$749$	−12.9083	−0.471660
$750$	0	0
$751$	−5.66947	−0.206882	−0.103441	−	0.994636i	$-0.532985\pi$
$751$	−5.66947	−0.206882	−0.103441	+	0.994636i	$0.532985\pi$
$752$	−0.908327	−0.0331233
$753$	5.09167	0.185551
$754$	−114.083	−4.15467
$755$	0	0
$756$	79.6611	2.89724
$757$	−23.0555	−0.837967	−0.418983	−	0.907994i	$-0.637613\pi$
$757$	−23.0555	−0.837967	−0.418983	+	0.907994i	$0.637613\pi$
$758$	57.1472	2.07568
$759$	−4.81665	−0.174833
$760$	0	0
$761$	−42.4222	−1.53780	−0.768902	−	0.639367i	$-0.779197\pi$
$761$	−42.4222	−1.53780	−0.768902	+	0.639367i	$0.779197\pi$
$762$	51.3583	1.86051
$763$	−28.0278	−1.01467
$764$	22.1194	0.800253
$765$	0	0
$766$	−49.8167	−1.79995
$767$	1.05551	0.0381124
$768$	23.3305	0.841868
$769$	−26.8167	−0.967033	−0.483517	−	0.875335i	$-0.660641\pi$
$769$	−26.8167	−0.967033	−0.483517	+	0.875335i	$0.660641\pi$
$770$	0	0
$771$	23.4500	0.844530
$772$	−4.00000	−0.143963
$773$	−22.1194	−0.795581	−0.397790	−	0.917476i	$-0.630223\pi$
$773$	−22.1194	−0.795581	−0.397790	+	0.917476i	$0.630223\pi$
$774$	21.6333	0.777593
$775$	0	0
$776$	35.0917	1.25972
$777$	−53.8444	−1.93166
$778$	27.6333	0.990702
$779$	−1.60555	−0.0575248
$780$	0	0
$781$	−4.60555	−0.164800
$782$	33.2750	1.18991
$783$	−55.5416	−1.98490
$784$	3.48612	0.124504
$785$	0	0
$786$	−2.72498	−0.0971968
$787$	−4.21110	−0.150110	−0.0750548	−	0.997179i	$-0.523913\pi$
$787$	−4.21110	−0.150110	−0.0750548	+	0.997179i	$0.523913\pi$
$788$	32.0278	1.14094
$789$	1.54163	0.0548836
$790$	0	0
$791$	46.5416	1.65483
$792$	3.90833	0.138876
$793$	14.5416	0.516389
$794$	58.2666	2.06780
$795$	0	0
$796$	−80.9638	−2.86969
$797$	−14.5139	−0.514108	−0.257054	−	0.966397i	$-0.582752\pi$
$797$	−14.5139	−0.514108	−0.257054	+	0.966397i	$0.582752\pi$
$798$	12.9083	0.456950
$799$	11.7250	0.414800
$800$	0	0
$801$	−6.90833	−0.244094
$802$	−62.6611	−2.21264
$803$	−2.90833	−0.102633
$804$	17.2111	0.606989
$805$	0	0
$806$	−48.4861	−1.70785
$807$	30.9083	1.08802
$808$	52.5416	1.84841
$809$	−3.63331	−0.127740	−0.0638701	−	0.997958i	$-0.520344\pi$
$809$	−3.63331	−0.127740	−0.0638701	+	0.997958i	$0.520344\pi$
$810$	0	0
$811$	−54.8722	−1.92682	−0.963411	−	0.268028i	$-0.913628\pi$
$811$	−54.8722	−1.92682	−0.963411	+	0.268028i	$0.913628\pi$
$812$	−140.808	−4.94140
$813$	−18.5139	−0.649310
$814$	−22.1194	−0.775286
$815$	0	0
$816$	1.54163	0.0539680
$817$	7.21110	0.252285
$818$	18.9083	0.661114
$819$	−28.0278	−0.979369
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	6.27502	0.218866
$823$	−10.4222	−0.363295	−0.181648	−	0.983364i	$-0.558143\pi$
$823$	−10.4222	−0.363295	−0.181648	+	0.983364i	$0.558143\pi$
$824$	−23.7250	−0.826499
$825$	0	0
$826$	2.09167	0.0727786
$827$	−7.81665	−0.271812	−0.135906	−	0.990722i	$-0.543394\pi$
$827$	−7.81665	−0.271812	−0.135906	+	0.990722i	$0.543394\pi$
$828$	15.9083	0.552853
$829$	−38.7527	−1.34594	−0.672969	−	0.739671i	$-0.734981\pi$
$829$	−38.7527	−1.34594	−0.672969	+	0.739671i	$0.734981\pi$
$830$	0	0
$831$	−28.1472	−0.976415
$832$	−64.0833	−2.22169
$833$	−45.0000	−1.55916
$834$	−24.6333	−0.852982
$835$	0	0
$836$	3.30278	0.114229
$837$	−23.6056	−0.815927
$838$	31.3305	1.08230
$839$	16.1194	0.556505	0.278252	−	0.960508i	$-0.410245\pi$
$839$	16.1194	0.556505	0.278252	+	0.960508i	$0.410245\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	9.90833	0.341463
$843$	29.7250	1.02378
$844$	83.3583	2.86931
$845$	0	0
$846$	9.00000	0.309426
$847$	4.30278	0.147845
$848$	0.697224	0.0239428
$849$	−3.51388	−0.120596
$850$	0	0
$851$	−35.5139	−1.21740
$852$	−19.8167	−0.678907
$853$	19.7250	0.675370	0.337685	−	0.941259i	$-0.390356\pi$
$853$	19.7250	0.675370	0.337685	+	0.941259i	$0.390356\pi$
$854$	28.8167	0.986086
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	15.0000	0.512092
$859$	48.6056	1.65840	0.829200	−	0.558952i	$-0.188796\pi$
$859$	48.6056	1.65840	0.829200	+	0.558952i	$0.188796\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	75.9916	2.58828
$863$	−19.6056	−0.667381	−0.333690	−	0.942683i	$-0.608294\pi$
$863$	−19.6056	−0.667381	−0.333690	+	0.942683i	$0.608294\pi$
$864$	−29.7250	−1.01126
$865$	0	0
$866$	11.5139	0.391258
$867$	2.24726	0.0763210
$868$	−59.8444	−2.03125
$869$	0.0916731	0.00310980
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−19.5416	−0.661763
$873$	−15.2389	−0.515757
$874$	8.51388	0.287986
$875$	0	0
$876$	−12.5139	−0.422805
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	47.6611	1.60848
$879$	−19.8167	−0.668399
$880$	0	0
$881$	−34.5416	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$881$	−34.5416	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$882$	−34.5416	−1.16308
$883$	−12.4500	−0.418975	−0.209487	−	0.977811i	$-0.567179\pi$
$883$	−12.4500	−0.418975	−0.209487	+	0.977811i	$0.567179\pi$
$884$	−64.5416	−2.17077
$885$	0	0
$886$	−3.21110	−0.107879
$887$	47.2389	1.58613	0.793063	−	0.609140i	$-0.208485\pi$
$887$	47.2389	1.58613	0.793063	+	0.609140i	$0.208485\pi$
$888$	−37.5416	−1.25981
$889$	−73.6611	−2.47051
$890$	0	0
$891$	3.39445	0.113718
$892$	68.1472	2.28174
$893$	3.00000	0.100391
$894$	−8.36669	−0.279824
$895$	0	0
$896$	−81.3583	−2.71799
$897$	24.0833	0.804117
$898$	95.5971	3.19012
$899$	41.7250	1.39161
$900$	0	0
$901$	−9.00000	−0.299833
$902$	−3.69722	−0.123104
$903$	40.4222	1.34517
$904$	32.4500	1.07927
$905$	0	0
$906$	62.4500	2.07476
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	17.5139	0.581218
$909$	−22.8167	−0.756781
$910$	0	0
$911$	−39.2111	−1.29912	−0.649561	−	0.760310i	$-0.725047\pi$
$911$	−39.2111	−1.29912	−0.649561	+	0.760310i	$0.725047\pi$
$912$	0.394449	0.0130615
$913$	−14.5139	−0.480339
$914$	−55.9638	−1.85112
$915$	0	0
$916$	45.3305	1.49776
$917$	3.90833	0.129064
$918$	−50.4500	−1.66510
$919$	41.2111	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$919$	41.2111	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$920$	0	0
$921$	7.93608	0.261503
$922$	40.9638	1.34907
$923$	23.0278	0.757968
$924$	18.5139	0.609062
$925$	0	0
$926$	60.3583	1.98350
$927$	10.3028	0.338388
$928$	52.5416	1.72476
$929$	46.3944	1.52215	0.761076	−	0.648662i	$-0.224671\pi$
$929$	46.3944	1.52215	0.761076	+	0.648662i	$0.224671\pi$
$930$	0	0
$931$	−11.5139	−0.377352
$932$	16.8167	0.550848
$933$	21.9083	0.717246
$934$	56.7250	1.85610
$935$	0	0
$936$	−19.5416	−0.638738
$937$	5.21110	0.170239	0.0851196	−	0.996371i	$-0.472873\pi$
$937$	5.21110	0.170239	0.0851196	+	0.996371i	$0.472873\pi$
$938$	−39.6333	−1.29407
$939$	−28.4222	−0.927524
$940$	0	0
$941$	52.3944	1.70801	0.854005	−	0.520265i	$-0.174167\pi$
$941$	52.3944	1.70801	0.854005	+	0.520265i	$0.174167\pi$
$942$	−14.3667	−0.468092
$943$	−5.93608	−0.193305
$944$	0.0639167	0.00208031
$945$	0	0
$946$	16.6056	0.539893
$947$	−36.6333	−1.19042	−0.595211	−	0.803569i	$-0.702932\pi$
$947$	−36.6333	−1.19042	−0.595211	+	0.803569i	$0.702932\pi$
$948$	0.394449	0.0128111
$949$	14.5416	0.472041
$950$	0	0
$951$	−12.9083	−0.418581
$952$	−50.4500	−1.63509
$953$	−49.2666	−1.59590	−0.797951	−	0.602722i	$-0.794083\pi$
$953$	−49.2666	−1.59590	−0.797951	+	0.602722i	$0.794083\pi$
$954$	−6.90833	−0.223665
$955$	0	0
$956$	−13.6056	−0.440035
$957$	−12.9083	−0.417267
$958$	−30.3583	−0.980832
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−13.2666	−0.427955
$962$	110.597	3.56580
$963$	3.90833	0.125944
$964$	−82.3583	−2.65258
$965$	0	0
$966$	47.7250	1.53553
$967$	4.09167	0.131579	0.0657897	−	0.997834i	$-0.479043\pi$
$967$	4.09167	0.131579	0.0657897	+	0.997834i	$0.479043\pi$
$968$	3.00000	0.0964237
$969$	−5.09167	−0.163568
$970$	0	0
$971$	−30.3583	−0.974244	−0.487122	−	0.873334i	$-0.661953\pi$
$971$	−30.3583	−0.974244	−0.487122	+	0.873334i	$0.661953\pi$
$972$	−40.9361	−1.31303
$973$	35.3305	1.13264
$974$	−23.5139	−0.753433
$975$	0	0
$976$	0.880571	0.0281864
$977$	−15.9722	−0.510997	−0.255499	−	0.966809i	$-0.582240\pi$
$977$	−15.9722	−0.510997	−0.255499	+	0.966809i	$0.582240\pi$
$978$	−17.0917	−0.546531
$979$	−5.30278	−0.169477
$980$	0	0
$981$	8.48612	0.270941
$982$	−55.7527	−1.77914
$983$	−48.8444	−1.55789	−0.778947	−	0.627089i	$-0.784246\pi$
$983$	−48.8444	−1.55789	−0.778947	+	0.627089i	$0.784246\pi$
$984$	−6.27502	−0.200040
$985$	0	0
$986$	89.1749	2.83991
$987$	16.8167	0.535280
$988$	−16.5139	−0.525376
$989$	26.6611	0.847773
$990$	0	0
$991$	−6.09167	−0.193508	−0.0967542	−	0.995308i	$-0.530846\pi$
$991$	−6.09167	−0.193508	−0.0967542	+	0.995308i	$0.530846\pi$
$992$	22.3305	0.708995
$993$	18.7527	0.595100
$994$	45.6333	1.44740
$995$	0	0
$996$	−62.4500	−1.97880
$997$	14.2750	0.452094	0.226047	−	0.974116i	$-0.427420\pi$
$997$	14.2750	0.452094	0.226047	+	0.974116i	$0.427420\pi$
$998$	−49.5416	−1.56821
$999$	53.8444	1.70356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.a.f.1.2	yes	2
3.2	odd	2		2475.2.a.o.1.1		2
4.3	odd	2		4400.2.a.bh.1.2		2
5.2	odd	4		275.2.b.c.199.4		4
5.3	odd	4		275.2.b.c.199.1		4
5.4	even	2		275.2.a.e.1.1	✓	2
11.10	odd	2		3025.2.a.h.1.1		2
15.2	even	4		2475.2.c.k.199.1		4
15.8	even	4		2475.2.c.k.199.4		4
15.14	odd	2		2475.2.a.t.1.2		2
20.3	even	4		4400.2.b.y.4049.3		4
20.7	even	4		4400.2.b.y.4049.2		4
20.19	odd	2		4400.2.a.bs.1.1		2
55.54	odd	2		3025.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	5.4	even	2
275.2.a.f.1.2	yes	2	1.1	even	1	trivial
275.2.b.c.199.1		4	5.3	odd	4
275.2.b.c.199.4		4	5.2	odd	4
2475.2.a.o.1.1		2	3.2	odd	2
2475.2.a.t.1.2		2	15.14	odd	2
2475.2.c.k.199.1		4	15.2	even	4
2475.2.c.k.199.4		4	15.8	even	4
3025.2.a.h.1.1		2	11.10	odd	2
3025.2.a.n.1.2		2	55.54	odd	2
4400.2.a.bh.1.2		2	4.3	odd	2
4400.2.a.bs.1.1		2	20.19	odd	2
4400.2.b.y.4049.2		4	20.7	even	4
4400.2.b.y.4049.3		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} -1.30278 q^{3} +3.30278 q^{4} -3.00000 q^{6} +4.30278 q^{7} +3.00000 q^{8} -1.30278 q^{9} -1.00000 q^{11} -4.30278 q^{12} +5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -3.00000 q^{18} -1.00000 q^{19} -5.60555 q^{21} -2.30278 q^{22} -3.69722 q^{23} -3.90833 q^{24} +11.5139 q^{26} +5.60555 q^{27} +14.2111 q^{28} -9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} +1.30278 q^{33} -9.00000 q^{34} -4.30278 q^{36} +9.60555 q^{37} -2.30278 q^{38} -6.51388 q^{39} +1.60555 q^{41} -12.9083 q^{42} -7.21110 q^{43} -3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} -0.394449 q^{48} +11.5139 q^{49} +5.09167 q^{51} +16.5139 q^{52} +2.30278 q^{53} +12.9083 q^{54} +12.9083 q^{56} +1.30278 q^{57} -22.8167 q^{58} +0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -5.60555 q^{63} -12.8167 q^{64} +3.00000 q^{66} -4.00000 q^{67} -12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} -3.90833 q^{72} +2.90833 q^{73} +22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -15.0000 q^{78} -0.0916731 q^{79} -3.39445 q^{81} +3.69722 q^{82} +14.5139 q^{83} -18.5139 q^{84} -16.6056 q^{86} +12.9083 q^{87} -3.00000 q^{88} +5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} +5.48612 q^{93} -6.90833 q^{94} +6.90833 q^{96} +11.6972 q^{97} +26.5139 q^{98} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{3} + 3 q^{4} - 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} - 2 q^{11} - 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} - 2 q^{19} - 4 q^{21} - q^{22} - 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 2 * q + q^2 + q^3 + 3 * q^4 - 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 - 2 * q^11 - 5 * q^12 + 10 * q^13 + 9 * q^14 - 3 * q^16 + 3 * q^17 - 6 * q^18 - 2 * q^19 - 4 * q^21 - q^22 - 11 * q^23 + 3 * q^24 + 5 * q^26 + 4 * q^27 + 14 * q^28 - 9 * q^29 + 6 * q^31 - 7 * q^32 - q^33 - 18 * q^34 - 5 * q^36 + 12 * q^37 - q^38 + 5 * q^39 - 4 * q^41 - 15 * q^42 - 3 * q^44 + q^46 - 6 * q^47 - 8 * q^48 + 5 * q^49 + 21 * q^51 + 15 * q^52 + q^53 + 15 * q^54 + 15 * q^56 - q^57 - 24 * q^58 - 14 * q^59 - 5 * q^61 - 23 * q^62 - 4 * q^63 - 4 * q^64 + 6 * q^66 - 8 * q^67 - 15 * q^68 - 12 * q^69 + 2 * q^71 + 3 * q^72 - 5 * q^73 + 19 * q^74 - 3 * q^76 - 5 * q^77 - 30 * q^78 - 11 * q^79 - 14 * q^81 + 11 * q^82 + 11 * q^83 - 19 * q^84 - 26 * q^86 + 15 * q^87 - 6 * q^88 + 7 * q^89 + 25 * q^91 - 10 * q^92 + 29 * q^93 - 3 * q^94 + 3 * q^96 + 27 * q^97 + 35 * q^98 - q^99

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